Abstract
Consider M as a 3-homogeneous manifold. In this paper, we are going to study the behavior of the first eigenvalue of p-Laplace operator in a case of Bianchi classes along the normalized Ricci flow also we will give some upper and lower bounds for a such eigenvalue.
Highlights
Over the last few years, studying the geometric flows, specially the Ricci flow have become a topic of active research in both mathematics and physics
A geometric flow is an evolution of a geometric structure under a differential equation related to a functional usually associated with a curvature in a manifold
Consider M as a manifold with Riemannian metric g0, the family g(t) of Riemannian metrics on M have been called as an un-normalized Ricci flow when it satisfies the equation d g(t) = −2Ric (g(t)) dt g(0) = g0, (1)
Summary
Over the last few years, studying the geometric flows, specially the Ricci flow have become a topic of active research in both mathematics and physics. Consider M as a manifold with Riemannian metric g0, the family g(t) of Riemannian metrics on M have been called as an un-normalized Ricci flow when it satisfies the equation d g(t) = −2Ric (g(t)) dt g(0) = g0, (1) One can consider the normalized Ricci flow as follow d Consider g(t) as a solution of the Ricci flow (1), the customary normalization on 3-manifolds is setting t g (t) = ψ(t)g(t), t = ψ(ν)dν, with 1 ∂ψ ψ ∂t
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